Question

Which of the following worked equations demonstrates that you can divide polynomials by recognizing division as the inverse operation of multiplication?(1 point)
Responses

8x2−4x+124x=(4x)(8x2−4x+12)

8x2−4x+124x=(−14x)(8x2−4x+12)

8x2−4x+124x=(14x)(8x2−4x+12)

8x2−4x+124x=(−4x)(8x2−4x+12)

Answer 1

To determine which equation demonstrates that you can divide polynomials by recognizing division as the inverse operation of multiplication, we need to identify whether each response properly represents the division operation as the multiplication of a factor by a polynomial.

In general, if \( A = B \cdot C \), then \( \frac{A}{B} = C \) as long as \( B \neq 0 \).

Let's analyze the given responses:

1. \( \frac{8x^2 - 4x + 12}{4x} = (4x)(8x^2 - 4x + 12) \)
- This equation suggests multiplying, not dividing.

2. \( \frac{8x^2 - 4x + 12}{4x} = (-\frac{1}{4}x)(8x^2 - 4x + 12) \)
- This is incorrect because it's also representing multiplication rather than division.

3. \( \frac{8x^2 - 4x + 12}{4x} = (\frac{1}{4}x)(8x^2 - 4x + 12) \)
- This option is valid rearrangement showing division correctly.

4. \( \frac{8x^2 - 4x + 12}{4x} = (-4x)(8x^2 - 4x + 12) \)
- Similar to others, this is also multiplication.

From the given options, the correct response that accurately illustrates division as the inverse of multiplication is:

**8x^2−4x+12 / 4x = (-\frac{1}{4}x)(8x^2−4x+12)**

This response expresses division while presenting multiplication as an inverse relationship.

So, the correct answer is the third option with correction of the sign:
**\( \frac{8x^2 - 4x + 12}{4x} = ( \frac{1}{4}x)(8x^2 - 4x + 12) \)**.